Geometry & Topology

Kähler–Ricci flow, Kähler–Einstein metric, and K–stability

Xiuxiong Chen, Song Sun, and Bing Wang

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Abstract

We prove the existence of a Kähler–Einstein metric on a K–stable Fano manifold using the recent compactness result on Kähler–Ricci flows. The key ingredient is an algebrogeometric description of the asymptotic behavior of Kähler–Ricci flow on Fano manifolds. This is in turn based on a general finite-dimensional discussion, which is interesting on its own and could potentially apply to other problems. As one application, we relate the asymptotics of the Calabi flow on a polarized Kähler manifold to K–stability, assuming bounds on geometry.

Article information

Source
Geom. Topol., Volume 22, Number 6 (2018), 3145-3173.

Dates
Received: 19 October 2015
Revised: 27 February 2018
Accepted: 27 March 2018
First available in Project Euclid: 29 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1538186735

Digital Object Identifier
doi:10.2140/gt.2018.22.3145

Mathematical Reviews number (MathSciNet)
MR3858762

Zentralblatt MATH identifier
06945124

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)
Secondary: 14J45: Fano varieties

Keywords
Kähler Ricci flow convergence uniqueness Fano manifold K–stability

Citation

Chen, Xiuxiong; Sun, Song; Wang, Bing. Kähler–Ricci flow, Kähler–Einstein metric, and K–stability. Geom. Topol. 22 (2018), no. 6, 3145--3173. doi:10.2140/gt.2018.22.3145. https://projecteuclid.org/euclid.gt/1538186735


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